Estimation of conditional ranks and tests of exogeneity in nonparametric nonseparable models

Abstract

Consider a nonparametric nonseparable regression model Y=g(Z,U), where g(Z,U) is increasing in U and U is uniform on [0,1]. We suppose that there exists an instrument W that is independent of U. The observable random variables are Y, Z and W, all one-dimensional. The purpose of this paper is twofold. First, we study the asymptotic properties of a kernel estimator of the distribution of V = F(Y|Z), which equals U when Z is exogenous. We show that this estimator converges to the uniform distribution at faster rate than the parametric n^{-1/2}-rate. Next, we construct test statistics for the hypothesis that Z is exogenous. The test statistics are based on the observation that Z is exogenous if and only if V is independent of W, and hence they do not require the estimation of the function g. The asymptotic properties of the proposed tests are proved, and a bootstrap approximation of the critical values of the tests is shown to work for finite samples via simulations. An empirical example using the U.K. Family Expenditure Survey is also given. (With Jean-Pierre Florens and Frédérique Fève)